n-Strongly Gorenstein Graded Modules

Let R be a graded ring and n ⩾ 1 an integer. We introduce and study n-strongly Gorenstein gr-projective, gr-injective and gr-flat modules. Some examples are given to show that n-strongly Gorenstein gr-injective (gr-projective, gr-flat, respectively) modules need not be m-strongly Gorenstein gr-injective (gr-projective, gr-flat, respectively) modules whenever n > m. Many properties of the n-strongly Gorenstein gr-injective and gr-flat modules are discussed, some known results are generalized. Then we investigate the relations between the graded and the ungraded n-strongly Gorenstein injective (or flat) modules. In addition, the connections between the n-strongly Gorenstein gr-projective, gr-injective and gr-flat modules are considered.